Final answer:
The time it takes for a pendulum to swing is directly proportional to the square root of its length. By setting up a proportion using the given values, we can find the time it would take for a longer pendulum. In this case, the pendulum would take approximately 0.74 seconds to swing once if it were extended to be 12.96cm long.
Step-by-step explanation:
To determine the relationship between the time it takes for a pendulum to swing and its length, we look at the given information. We know that the time t is directly proportional to the square root of the length l of the pendulum. This can be written as t = k√l, where k is the constant of proportionality.
Using the given values, we can set up a proportion. Let t1 and l1 be the time and length of the first pendulum, and t2 and l2 be the time and length of the second pendulum. We have t1 = k√l1 and t2 = k√l2.
Plugging in the given values, we have 0.65 = k√10.24 and we want to find t2 when l2 = 12.96. We can solve for k by rearranging the equation to k = t1 / √l1 and substitute it into the second equation to find t2 = (t1 / √l1)√l2. Plugging in the values, we have t2 ≈ (0.65 / √10.24)√12.96, which is approximately equal to 0.74 seconds.